Hypergeometric Polynomials, Hyperharmonic Discrete and Continuous Expansions: Evaluations, Interconnections, Extensions

The important mathematical subject of special functions and orthogonal polynomials found in the last decades a systematization regarding those of hypergeometric type. The growth of these developments are due to interconnections with quantum angular momentum theory which is basic to that of spin-networks, of recent relevance in various branches of physics. Here we consider their power as providing expansion basis sets such as specifically are needed in chemistry to represents potential energy surfaces, the achievements being discussed and illustrated. A novel visualization of key members of the polynomial sets attributes a central role to the Kravchuk polynomials: its relationship with Wigner’s rotation matrix elements are here emphasized and taken as exemplary for computational and analytical features. The sets are considered regarding progress on the formulation of a discretization technique, the hyperquantization, which allows to efficiently deal with physical problems where quantum mechanical operators act on continuous manifolds, to yield discrete grids suitable for computation of matrix elements without need of multidimensional integration.